find the principal unit normal vector to the curve at the specific value of the parameter.

$\displaystyle r(t) = ti + \frac {1}{2}t^2j, t = 2$

i need to use this formula

$\displaystyle T(t) = \frac {r'(t)}{ || r'(t) ||} $

so, to find r'(t)

$\displaystyle r'(t) = i + t j $

so, now to find the magnitude: simplified

$\displaystyle || r'(t) || = \sqrt {1 + t^2} $

puting them together

$\displaystyle T(t) = \frac {i + t j} {\sqrt {1 + t^2}} $

From here is where i get lost

I need to find T'(t)

I can seem to follow the solutions manual. I think they are using the quiotient rule due to the denominator having the 3/2

fraction. But if they are i still cant see how they are doing it.

Can anyone help me work out the next step for T'(t)?

$\displaystyle T'(t) = \frac { -t}{(t^2 + 1)^{3/2}}i + \frac {1}{ (t^2 + 1) ^{3/2}} j$

Thank you so much!!!